This exercise, we refer back to exercise 16 which was the mark of chain, which had to do with how Maney umbrellas Sara head at home and in exercise 16. We had come up with a single step transition matrix here and now we're asked in part a that if Sarah has two umbrellas at home on Sunday night, what is the probability of having exactly two umbrellas at home on the next Friday night? So that is exactly five days later, five working days later and say and is equal to five. And so we're looking for the five step transition probability. I'm going from two to you, too. And, as usual, the way to find the transition probabilities for five steps is to take the single step transition matrix and raise it to the explode and five so he could do that in software. And so then this is equal to the entry to to in that matrix. Now we do have to be careful, because when we came up with this transition matrix and exercise 16 I had indexed the rows and columns as follows. Just so, it was easier to keep track of how many umbrellas we're at home. So here, when I say entry to to actually mean this entry right here, which is entry, actually entry 33 in in conventional matrix terminology and in the five step transition matrix, that number happens to be 0.274 So that's that's sort of the first question of part. And the second question is what is the probability by a Friday evening that there will be at least two umbrellas at the house? So this time we're looking for the five step probability going from 2 to 2, unless the five step probability of going from 2 to 3, as well as from 2 to 4. And so this is equal to zero point 274 less zero point 279 plus 0.2 to 1, which comes out to 0.775 now for Part B were given that there are two umbrellas on Sunday night at home. And what are the chances that there will be zero umbrellas to take to work on the following Thursday morning? So that is the same is asking the question. What is the probability that there will be zero umbrellas at home on Wednesday night after work. So that is three steps from Sunday night to Wednesday night. That means we're looking for the three step probability. I'm going from two umbrellas at home to know umbrellas at home. So this time, of course, we calculate we calculate P three, which is the three step transition matrix or the Markov chain. And then we look at Entry 31 which corresponds to going from 2 to 0. Just to make that clear would be going for entry three one, which would be this entry. Except that this is the single step transition matrix. So that would be the probability of going from two 20 umbrellas. But of course these air actually really three and call him one, and this comes out to 0.38 in last report. See, we're told that Sarah has two umbrellas at home at the start of the week and were asked what the expected number of umbrellas you'll have at home at the end of Monday and at the end of Tuesday is so when we look at the transition matrix, So road Rule three, which is corresponds to having two umbrellas. It is thebe probability distribution for how many umbrellas are going to be at home after the next transition. So if we have two umbrellas at home, the probability of having zero umbrellas at home in one step is zero. The probability of having one umbrella at home in one step is 0.14 and so on. So this role was their probability distribution for how many umbrellas we're going to have at home after one transition. And so the expected value of a random variable is the sum of X times probability of X, And the summation is overall possible values of X. And in this case, X goes from 0 to 4. That's the number of umbrellas. It is the state space for this Markov chain. So looking at that row of the transition matrix, the probability of getting zero is zero. The probability of one is 0.14 Probability of getting to is 0.62 probability, so that should be multiplied by two and then three, and probability of three is 0.24 and the probability of getting four is zero, and that comes out to zero. That comes out to 2.1, so we could call that sort of the expected number of umbrellas, um, after one transition, which happens to be Monday night. So that's the answer to the first question. And then the second question is, what is the expected number of umbrellas tohave on Tuesday night? If we start with two on Sunday night, so this time we proceed in a similar manner, except we need to use the two step transition matrix. So we calculate this, so I won't write it all out. I'll just give the pertinent row. 0.196 zero point 1736 zero point 4516 0.2976 and 0.576 So if we currently have two umbrellas at home in two steps, our probability distribution for the number of umbrellas at home looks like this role. And so that is our probability distribution for how many umbrellas we're going to have at home on Tuesday night. So what is the expected number of umbrellas at home on Tuesday night? So it zero times 0.196 It's one times 0.1736 two times 0.4516 and so on and this comes out to 2.2.